In nonparametric regression problems involving multiple predictors, there is typically interest in estimating an anisotropic multivariate regression surface in the important predictors while discarding the unimportant ones. Our focus is on defining a Bayesian procedure that leads to the minimax optimal rate of posterior contraction (up to a log factor) adapting to the unknown dimension and anisotropic smoothness of the true surface. We propose such an approach based on a Gaussian process prior with dimension-specific scalings, which are assigned carefully-chosen hyperpriors. We additionally show that using a homogenous Gaussian process with a single bandwidth leads to a sub-optimal rate in anisotropic cases.
We consider nonparametric Bayesian estimation inference using a rescaled smooth Gaussian field as a prior for a multidimensional function. The rescaling is achieved using a Gamma variable and the procedure can be viewed as choosing an inverse Gamma bandwidth. The procedure is studied from a frequentist perspective in three statistical settings involving replicated observations (density estimation, regression and classification). We prove that the resulting posterior distribution shrinks to the distribution that generates the data at a speed which is minimax-optimal up to a logarithmic factor, …
University of Washington（华盛顿大学）；University of Pennsylvania（宾夕法尼亚大学）
We study the clustering task under anisotropic Gaussian Mixture Models where the covariance matrices from different clusters are unknown and are not necessarily the identical matrix. We characterize the dependence of signal-to-noise ratios on the cluster centers and covariance matrices and obtain the minimax lower bound for the clustering problem. In addition, we propose a computationally feasible procedure and prove it achieves the optimal rate within a few iterations. The proposed procedure is a hard EM type algorithm, and it can also be seen as a variant of the Lloyd's algorithm that is adj…
We study the supervised clustering problem under the two-component anisotropic Gaussian mixture model in high dimensions and in the non-asymptotic setting. We first derive a lower and a matching upper bound for the minimax risk of clustering in this framework. We also show that in the high-dimensional regime, the linear discriminant analysis (LDA) classifier turns out to be sub-optimal in the minimax sense. Next, we characterize precisely the risk of $\ell_2$-regularized supervised least squares classifiers. We deduce the fact that the interpolating solution may outperform the regularized clas…
This work is intended as a contribution to a wavelet-based adaptive estimator of the memory parameter in the classical semi-parametric framework for Gaussian stationary processes. In particular we introduce and develop the choice of a data-driven optimal bandwidth. Moreover, we establish a central limit theorem for the estimator of the memory parameter with the minimax rate of convergence (up to a logarithm factor). The quality of the estimators are attested by simulations.
We present a new, elementary way to obtain axially symmetric Gaussian processes on the sphere, in order to accommodate for the directional anisotropy of global climate data in geostatistical analysis.
A majority of experimental disciplines face the challenge of exploring large and high-dimensional parameter spaces in search of new scientific discoveries. Materials science is no exception; the wide variety of synthesis, processing, and environmental conditions that influence material properties gives rise to particularly vast parameter spaces. Recent advances have led to an increase in efficiency of materials discovery by increasingly automating the exploration processes. Methods for autonomous experimentation have become more sophisticated recently, allowing for multi-dimensional parameter …
It is well known that the number of modes of a kernel density estimator is monotone nonincreasing in the bandwidth if the kernel is a Gaussian density. There is numerical evidence of nonmonotonicity in the case of some non-Gaussian kernels, but little additional information is available. The present paper provides theoretical and numerical descriptions of the extent to which the number of modes is a nonmonotone function of bandwidth in the case of general compactly supported densities. Our results address popular kernels used in practice, for example, the Epanechnikov, biweight and triweight k…
This paper studies polar sets of anisotropic Gaussian random fields, i.e. sets which a Gaussian random field does not hit almost surely. The main assumptions are that the eigenvalues of the covariance matrix are bounded from below and that the canonical metric associated with the Gaussian random field is dominated by an anisotropic metric. We deduce an upper bound for the hitting probabilities and conclude that sets with small Hausdorff dimension are polar. Moreover, the results allow for a translation of the Gaussian random field by a random field, that is independent of the Gaussian random f…
We consider the solution X = (Xt) t$\ge$0 of a multivariate stochastic differential equation with Levy-type jumps and with unique invariant probability measure with density $μ$. We assume that a continuous record of observations X T = (Xt) 0$\le$t$\le$T is available. In the case without jumps, Reiss and Dalalyan (2007) and Strauch (2018) have found convergence rates of invariant density estimators, under respectively isotropic and anisotropic H{ö}lder smoothness constraints, which are considerably faster than those known from standard multivariate density estimation. We extend the previous wor…
We estimate the anisotropic index of an anisotropic fractional Brownian field. For all directions, we give a convergent estimator of the value of the anisotropic index in this direction, based on generalized quadratic variations. We also prove a central limit theorem. First we present a result of identification that relies on the asymptotic behavior of the spectral density of a process. Then, we define Radon transforms of the anisotropic fractional Brownian field and prove that these processes admit a spectral density satisfying the previous assumptions. Finally we use simulated fields to test…
We look into the minimax results for the anisotropic two-dimensional functional deconvolution model with the two-parameter fractional Gaussian noise. We derive the lower bounds for the $L^p$-risk, $1 \leq p < \infty$, and taking advantage of the Riesz poly-potential, we apply a wavelet-vaguelette expansion to de-correlate the anisotropic fractional Gaussian noise. We construct an adaptive wavelet hard-thresholding estimator that attains asymptotically quasi-optimal convergence rates in a wide range of Besov balls. Such convergence rates depend on a delicate balance between the parameters of th…
We address the problem of density estimation with $\mathbb{L}_s$-loss by selection of kernel estimators. We develop a selection procedure and derive corresponding $\mathbb{L}_s$-risk oracle inequalities. It is shown that the proposed selection rule leads to the estimator being minimax adaptive over a scale of the anisotropic Nikol'skii classes. The main technical tools used in our derivations are uniform bounds on the $\mathbb{L}_s$-norms of empirical processes developed recently by Goldenshluger and Lepski [Ann. Probab. (2011), to appear].
We address the problem of adaptive minimax estimation in white gaussian noise model under $L_p$--loss, $1\leq p\leq\infty,$ on the anisotropic Nikolskii classes. We present the estimation procedure based on a new data-driven selection scheme from the family of kernel estimators with varying bandwidths. For proposed estimator we establish so-called Lp-norm oracle inequality and use it for deriving minimax adaptive results. We prove the existence of rate-adaptive estimators and fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm inde…
CREST, ENSAE, Institut Polytechnique de Paris（巴黎理工学院ENSAE CREST）；University of California, Berkeley（加州大学伯克利分校）
Assume that $X_{1}, \ldots, X_{N}$ is an $\varepsilon$-contaminated sample of $N$ independent Gaussian vectors in $\mathbb{R}^d$ with mean $μ$ and covariance $Σ$. In the strong $\varepsilon$-contamination model we assume that the adversary replaced an $\varepsilon$ fraction of vectors in the original Gaussian sample by any other vectors. We show that there is an estimator $\widehat μ$ of the mean satisfying, with probability at least $1 - δ$, a bound of the form \[ \|\widehatμ - μ\|_2 \le c\left(\sqrt{\frac{\operatorname{Tr}(Σ)}{N}} + \sqrt{\frac{\|Σ\|\log(1/δ)}{N}} + \varepsilon\sqrt{\|Σ\|}\r…
Anisotropic functional deconvolution model is investigated in the bivariate case under long-memory errors when the design points $t_i$, $i=1, 2, \cdots, N$, and $x_l$, $l=1, 2, \cdots, M$, are irregular and follow known densities $h_1$, $h_2$, respectively. In particular, we focus on the case when the densities $h_1$ and $h_2$ have singularities, but $1/h_1$ and $1/h_2$ are still integrable on $[0, 1]$. Under both Gaussian and sub-Gaussian errors, we construct an adaptive wavelet estimator that attains asymptotically near-optimal convergence rates that deteriorate as long-memory strengthens. T…
